Molecular mechanics is the name given to a widely used calculational method for providing accurate structures and energies for molecules. It has been long known that by employing the fundamentals of quantum mechanics, very accurate models of molecular structure could be achieved. Unfortunately, once the model size exceeded 15 or 20 atoms, the calculation became so time consuming as to be practically unsolvable. To overcome this problem, various models for the interaction of groups of atoms in molecules have been derived which provide reasonable approximations of interatomic relationships and energy distributions. A text which describes the field of molecular mechanics is entitled, appropriately, "Molecular Mechanics" by Burkert and Allinger, ACS Monograph 177, American Chemical Society, Washington, D.C. 1982.
The ability to calculate the structure and energies of a molecule are important for several reasons. As is well-known to chemists, the shape of a molecule is directly related to its properties. For instance, biologically active molecules are effective because they fit into certain openings in proteins. If it is desired to interfere with a natural action, a molecule must be designed which fits into that same opening and acts to inhibit the molecules action. This gives rise to the need to know the detailed structure for molecules, their bond distances, angles, etc.
While there are a number of methods for experimentally determining molecular structures, (e.g., X-Ray crystallography,) they first require the availability or synthesis of the substance, then the growing of a suitable crystal, then the accumulation of the X-Ray data and finally, computer processing of the data to obtain the structure. This process is time consuming and expensive at best, and difficulties at any step many times make it impossible.
Molecular Mechanics requires that the internal force fields in the molecule be approximated. For the purposes of the model, each atom is normally represented as a point and is connected to other atoms by bonds equivalent to springs. If a bond between two atoms (diatomic) is either stretched or compressed from an equilibrium distance, the bond is considered to increase in potential energy. Forces which act along covalent bonds and attempt to restore the equilibrium bond length are a component of the "valence force field".
In the case of a triatomic molecular model, (includes three atoms and has two bonds), the angle between the bonds is known as the bond angle and may also be stressed from its equilibrium point by the molecular structure. Both the diatomic and triatomic models naturally have all atoms in a single plane. However, when a fourth atom is introduced, there can be a torsion or dihedral angle between two planes, each one containing three of the atoms. Additionally, there are forces extant between atoms which are not directly bonded, which forces also come into play in the consideration of molecular structures.
It is the sum of all four of these potential energies, i.e., stretching energy, bending angle energy, torsion angle energy and non-bonded energy which comprise the steric energy of a molecule. Each of these individual energies will be described in further detail hereinbelow. Suffice it to say at this juncture that the basic idea is that bonds in a molecule have "natural" lengths and interbond angles, and molecules adjust their geometry so as to accommodate these values in most simple cases. In more strained systems, the molecules deform in predictable ways with strain energies that can be accurately calculated. It is clear, however, that there are certain preferred molecular structures which exhibit minimal steric energies and are most stable and most likely to exist. From a definitional point of view, an energy minimum in a molecule is referred to as a "conformer" or a "conformation". Butane, for instance, has three conformers which correspond to three energy minima. With more complicated molecules, there will, in general, be a large number of energy minima of different steric energies.
As aforestated, the molecular mechanics model of two atoms joined by a valence bond may be thought of as two points joined by a spring which has a natural value and which may be either compressed or extended. The angle of joinder between bonds in a triatomic species is similar in function to the bond "spring" in that it has a natural or unstressed angle and may be stressed to either compact or expand depending upon the molecular structure. The energy models of these atomic interrelationships are shown in the upper right hand corners of FIGS. 1 and 2 respectively.
In FIG. 1, the bond stretching energy function is shown as follows: EQU E.sub.s =k.sub.r (r-r.sub.o).sup.2
where
r=actual bond distance PA1 r.sub.o =nonstressed bond distance PA1 k.sub.r =stretching force constant PA1 .theta.=actual angle between bonds PA1 .theta..sub.o =nonstressed bond angle PA1 k.sub..theta. =angle bending force constant PA1 w=torsion angle PA1 k.sub.w =torsional constant
The function E.sub.s (i.e., the variation of bond energy with bond distance) is charted by dotted line 10 in FIG. 1. The actual variation of bond energy with bond stretching is shown by curve 12 which varies from the model by indicating that if the bond distance expands too far, the bond breaks and there is no further change in energy. Since that is not of practical concern, the model serves well to define the relationship. It is also to be noted that at distance r.sub.o, the energy is at a minimum. Thus, r.sub.o is defined as the equilibrium bond distance.
Turning to FIG. 2, a similar function is shown for angle bending as follows: ##EQU1## where
It will be noted that the function which defines E.sub..theta. describes a parabola which has a minimum at .theta..sub.o (the equilibrium angle).
With four or more atoms, one of the atoms (in this instance atom D in FIG. 3) may be out of the plane of the other three atoms A, B and C. In such a case, there is a torsion or dihedral force exerted on the molecular structure. The energy function which expresses the torsional relationship is a cosine function which is expressed as follows: EQU E.sub.t =k.sub.w (1-Cos(3w))
where
The chart in FIG. 3 shows a plot of the torsion energy function E.sub.t as torsion angle .theta. varies from 0.degree. to 360.degree..
The last energy function employed in conventional mechanics is that which results from non-bonded interactions between atoms, (i.e., as shown in FIG. 3, it is the interaction between atoms A and D). As aforestated, all of these functions are models of the actual relationships and, at best, are approximations. This is more so with respect to non-bonded interactions than with the other energy equations. The function shown in FIG. 4 for non-bonded energy is just one of a number which may be used. In the equation shown in FIG. 4, epsilon defines the depth of the potential well, r.sub.o the minimum energy distance between the two atoms and r the actual distance between the two atoms. A plot of the equation of FIG. 4 is shown to the left of the equation.
The constants for each of the equations shown in FIGS. 1-4 may be experimentally determined, and some are shown in the aforementioned text by Molecular Mechanics by Burkert et al. (e.g., see pp 39 & 40) For instance, the stretching force constant k.sub.r may be determined by an analysis of a molecule's infrared spectra because the vibration frequencies which give rise to the IR bands are related to the molecule's geometry and force constants. The other constants, i.e., angle bending force constant k.sub..theta. and torsion bending constant k.sub.w etc., are similarly determined experimentally.
Conventional molecular mechanics programs require that the user initially input a trial molecular structure including the types of atoms, position coordinates that define their spatial positions, and the bond connections. These initial structure are usually guessed. Once the computer calculates the necessary internal coordinate bond distances, angles, etc., it then proceeds to calculate the steric energy based upon a summation of the energies found from the equations shown in FIGS. 1-4 or through the use of other equivalent equations. Once the steric energy for the trial molecular structure is found, the program then tries to improve the structure by moving the atoms so that the calculated steric energy of the molecule decreases.
There are a number of prior art techniques for optimizing such atomic movements (e.g., "steepest descent"). In essence, each method involves an iterative, over and over, movement of the atoms around their coordinates to determine the direction of atomic movement which leads to an energy minima. In complex molecular structures, this will generally lead to a local minimum, but not necessarily the best minimum (global). This is especially true when complex molecules are being analyzed which have a great number of regional minima.
Thus, the results depend on both the force-field calculations and, to a great extent, the intuition of the person doing the calculation who chose which starting geometries were used.
In attempting to find all the conformations (and therefore also the lowest energy conformation--the global minimum), computer programs exist in the prior art which systematically step through different values of a dihedral angle in the starting structure. For instance, see Wiberg, K. B. and Boyd, R. H. "Journal of the American Chemical Society", 1972, Vol. 94, at page 8426. While this procedure works for small systems, with increasing complexity it is possible to miss the best structure because the step size is too large or because the initial structures for some angles refines to minima which are not "global minima" even for the particularly selected dihedral angle. The odds of finding the global minimum can be improved by stepping through all combinations of values of several dihedral angles; however, the person using the program must decide which angles to vary and by what steps. Too few angles or steps which are too large rapidly increase the chances of missing the best structure.
As above stated, computer programs are available which carry out aspects of the above recited molecular mechanics analytical steps. Some of those programs are made generally available through "Quantum Chemistry Program Exchange", Indiana University, Chemistry Building 204, Bloomington, Indiana 47401. One such program is MM2 by N. L. Allinger. Another program, i.e. MMP2, a program for general molecular mechanics calculations, is available from Molecular Design Limited, 1122B Street, Hayward, California 94541. As aforestated, however, those programs suffer from the drawback that the user is not aware of whether all of the minima have been discovered in any particular atomic arrangement, and, more particularly, whether the molecular structure which gives the lowest steric energy (and is thus most stable) has been found.
It is, therefore, an object of this invention to provide a method for molecular mechanics calculations which has a high probability of finding all of the steric energy minima for a particular molecular structure.
It is still another object of this invention to provide a molecular mechanics calculation method wherein a high probability is present of finding molecular structures which exhibits the best global steric energy minima.